Chapter 2.4, Problem 36E

To determine

To find: the solutions for the given system of equations.

Answer to Problem 36E

  $x=0.4,y=-2.1\text{ and }z=3.5$

Explanation of Solution

Given information:

The given system of equations is as follows:

  $x-2y=4.6$... (1)

  $y-z=-5.6$... (2)

  $x+y+z=1.8$... (3)

Calculation:

Rearrange equation (1) for x as shown below.

  $x=4.6+2y$... (4)

Rearrange equation (2) for y as shown below.

  $y=-5.6+z$... (5)

Put the values of x and y in equation (3) and evaluate as follows:

  $\begin{array}{c}x+y+z=1.8\\ \left(4.6+2y\right)+y+z=1.8\left[\text{Substitute value of }x\right]\\ 4.6+2y+y+z=1.8\\ 4.6+3y+z=1.8\\ 4.6+3\left(-5.6+z\right)+z=1.8\left[\text{Substitute value of }y\right]\\ 4.6-16.8+3z+z=1.8\\ -12.2+4z=1.8\\ 4z=1.8+12.2\\ 4z=14\\ z=\frac{14}{4}\\ =3.5\end{array}$

Put the values of z in equation (5) and evaluate as follows:

  $\begin{array}{c}y=-5.6+3.5\\ =-2.1\end{array}$

Put the values of y in equation (4) and evaluate as follows:

  $\begin{array}{c}x=4.6+2\left(-2.1\right)\\ =4.6-4.2\\ =0.4\end{array}$

Thus, the solution $\left(x,y,z\right)$ is $\left(0.4,-2.1,3.5\right)$ .